Kibble–Zurek mechanism in colloidal monolayers SEE COMMENTARY
Sven Deutschländer, Patrick Dillmann, Georg Maret, and Peter Keim1
Department of Physics, University of Konstanz, 78464 Konstanz, Germany
Edited by Tom C. Lubensky, University of Pennsylvania, Philadelphia, PA, and approved March 27, 2015 (received for review January 13, 2015) – ξ The Kibble Zurek mechanism describes the evolution of topolog- called the Ginzburg temperature TG, and the length scale d of ical defect structures like domain walls, strings, and monopoles the initial (proto)domains is supposed to be equal to the corre- ξ = ξð Þ when a system is driven through a second-order phase transition. lation length at that temperature: d TG (2). The model is used on very different scales like the Higgs field in The geometry of the defect network that separates the un- the early universe or quantum fluids in condensed matter systems. correlated domains is given by the topology of the manifold of A defect structure naturally arises during cooling if separated re- degenerated states that can exist in the low symmetry phase. gions are too far apart to communicate (e.g., about their orienta- Thus, it depends strongly on the dimensionality of the system D tion or phase) due to finite signal velocity. This lack of causality and on the dimension N of the order parameter itself. Regarding results in separated domains with different (degenerated) locally the square root of Eq. 2, the expectation value of a one-com- broken symmetry. Within this picture, we investigate the nonequi- ponent order parameter (N = 1) can only take two different low librium dynamics in a condensed matter analog, a 2D ensemble of symmetry values hϕi =±ηðTÞ (e.g., the magnetization in a 2D or colloidal particles. In equilibrium, it obeys the so-called Kosterlitz– 3D Ising model): the manifold of the possible states is discon- Thouless–Halperin–Nelson–Young (KTHNY) melting scenario with nected. This topological constraint has a crucial effect if one continuous (second order-like) phase transitions. The ensemble is considers a mesh of symmetry broken domains where hϕi is exposed to a set of finite cooling rates covering roughly three chosen randomly as either +η or −η. If two neighboring (but orders of magnitude. Along this process, we analyze the defect uncorrelated) domains have the same expectation value of hϕi, and domain structure quantitatively via video microscopy and de- they can merge. In contrast, domains with an opposite expecta- termine the scaling of the corresponding length scales as a func- tion value will be separated by a domain wall in 3D (or a domain tion of the cooling rate. We indeed observe the scaling predicted line in 2D). At its center, the domain wall attains a value of – by the Kibble Zurek mechanism for the KTHNY universality class. hϕi = 0, providing a continuous crossover of the expectation value between the domains (Fig. 1A). Consider now N = 2: hϕi Kibble–Zurek mechanism | nonequilibrium dynamics | spontaneous hϕi2 = hϕ i2 + hϕ i2 = η2ð Þ can take any value on a circle, e.g., x y T symmetry breaking | KTHNY theory | colloids (all of the order parameter values that are lying on the minimum circle of the sombrero are degenerated). Because the manifold n the formalism of gauge theory with spontaneously broken of possible low symmetry states is now connected, hϕi can vary Isymmetry, Zel’dovich et al. and Kibble postulated a cosmo- smoothly along a path (Fig. 1B). In a network of symmetry logical phase transition during the cooling down of the early broken domains in two dimensions, at least three domains (in universe. This transition leads to degenerated states of vacua Fig. 1B separated by dashed lines) meet at a mutual edge. On a below a critical temperature, separated or dispersed by defect closed path around the edge, the expectation value hϕi might be structures as domain walls, strings, or monopoles (1–3). In the either constant along the path (for a global, uniform ϕ) but can course of the transition, the vacuum can be described via an also vary by a multiple of 2π (in analogy to the winding numbers N-component, scalar order parameter ϕ (known as the Higgs in liquid crystals). In the first case, the closed path can be re- field) underlying an effective potential duced to a point with hϕi ≠ 0, and no defect is built. If the path is shrunk in the second case, the field eventually has to attain = ϕ2 + ϕ2 − η2 2 [1] V a b 0 , Significance η where a is temperature dependent, b is a constant, and 0 is the hϕi = modulus of at T 0. For high temperatures, V has a single Spontaneous symmetry breaking describes a variety of trans- ϕ = minimum at 0 (high symmetry) but develops a minimum formations from high- to low-temperature phases and applies to “ ” landscape of degenerated vacua below a critical temperature cosmological concepts as well as atomic systems. T. W. B. Kibble = Tc (e.g., the so-called sombrero shape for N 2). Cooling down suggested defect structures (domain walls, strings, and mono- from the high symmetry phase, the system undergoes a phase poles) to appear during the expansion and cooling of the early transition at Tc into an ordered (low symmetry) phase with non- universe. The lack of such defects within the visible horizon of hϕi < zero . For T Tc it holds the universe mainly motivated inflationary Big Bang theories. W. H. Zurek pointed out that the same principles are relevant hϕi2 = η2 − 2 2 = η2ð Þ [2] 0 1 T Tc T . within the laboratory when a system obeying a second-order phase transition is cooled at finite rates into the low symmetry Caused by thermal fluctuations, one can expect that below Tc, phase. Using a colloidal system, we visualize the Kibble–Zurek hϕi takes different nonzero values in regions that are not con- mechanism on single particle level and clarify its nature in the PHYSICS nected by causality. The question now arising concerns the de- background of an established microscopic melting formalism. ξ termination of the typical length scale d of these regions and ξ their separation. For a finite cooling rate, d is limited by the Author contributions: P.K. designed research; S.D. and P.K. performed research; P.D. speed of propagating information, which is given by the finite contributed new reagents/analytic tools; S.D. analyzed data; and S.D., G.M., and P.K. speed of light defining an ultimate event horizon. Independent of wrote the paper. the nature of the limiting causality, Kibble argued that as long as The authors declare no conflict of interest. the difference in free energy ΔF (of a certain system volume) This article is a PNAS Direct Submission. between its high symmetry state hϕi = 0 and a possible finite Freely available online through the PNAS open access option. value of hϕi just below Tc is less than kBT, the volume can jump See Commentary on page 6780. 1 between both phases. The temperature at which ΔF = kBT is To whom correspondence should be addressed. Email: [email protected].
www.pnas.org/cgi/doi/10.1073/pnas.1500763112 PNAS | June 2, 2015 | vol. 112 | no. 22 | 6925–6930 Downloaded by guest on September 29, 2021 AB ^t = τð^tÞ. [3] <φ> = +η 2 2 2 <φx> +<φy> = η path The frozen-out correlation length ^ξ is then set at the temperature ^e of the corresponding freeze-out time: ^ξ = ξð^eÞ = ξð^tÞ. For a lin- ear temperature quench <φ> = 0 ‘path’ e = ðT − TcÞ=Tc = t τq, [4]
=ð +μÞ with the quench time scale τ , one observes ^t = ðτ τμÞ1 1 and <φ> = 0 q 0 q <φ> = -η ^ ^ ν=ð1+μÞ ξ = ξðtÞ = ξ τq τ0 . [5] Fig. 1. Emergence of defects in the Higgs field that is illustrated with red 0 vectors (shown in 2D for simplicity). (A) For N = 1 and D = 3, domain walls can D = B N = D = For the Ginzburg–Landau model (ν = 1=2, μ = 1), one finds the appear (strings for 2). ( )For 2 and 3, nontrivial topologies are ^ 1=4 strings (monopoles for D = 2). The defects are regions where the order pa- scaling ξ ∼ τq , whereas a renormalization group correction ^ 1=3 rameter ϕ retains the high symmetry phase (hϕi = 0) to moderate between (ν = 2=3) leads to ξ ∼ τq (4, 5). different degenerated orientations of the symmetry broken field. A frequently used approximation is that when the adiabatic regime ends at ^t before the transition, the correlation length cannot follow the critical behavior until τ again exceeds the time hϕi = 0 within the path, and one remains with a monopole for t when T is passed. Given a symmetric divergence of τ around = = c ^ D 2 or a string for D 3 (2, 3). A condensed matter analog Tc, this is the time t after the transition. The period in between is would be a vortex of normal fluid with quantized circulation in known as the impulse regime, in which the correlation length is superfluid helium. For N = 3 and D = 3, four domains can meet at assumed not to evolve further. A recent analytical investigation, a mutual point, and the degenerated solutions of the low tem- however, suggests that in this period, the system falls into a re- hϕi2 = hϕ i2 + hϕ i2 + hϕ i2 perature phase lie on a sphere: x y z . If the gime of critical coarse graining (6). There, the typical length field now again varies circularly on a spherical path (all field scale of correlated domains continues to grow because local arrows point radially outward), a shrinking of this sphere leads to fluctuations are still allowed, and the system is out of equilib- a monopole in three dimensions (2, 3). rium. On the other side, numerical studies in which dissipative Zurek extended Kibble’s predictions and transferred his con- contributions and cooling rates were alternatively varied before siderations to quantum condensed matter systems. He suggested and after the transition indicate that the final length scale of the that 4He should intrinsically develop a defect structure when defect and domain network is entirely determined after the transition (7). Several efforts have been made to provide ex- quenched from the normal to the superfluid phase (4, 5). For – superfluid 4He, the order parameter ψ = jψjexpðiΘÞ is complex perimental verification of the Kibble Zurek mechanism in a with two independent components: magnitude jψj and phase Θ variety of systems, e.g., in liquid crystals (8) (the transition is 2 weakly first order but the defect network can easily observed (the superfluid density is given by jψj ). A nontrivial, static so- with cross polarization microscopy), superfluid 3He (9), super- lution of the equation of state with a Ginzburg–Landau potential ψ = ψ ð Þ ð φÞ φ conducting systems (10), convective, intrinsically out of equilib- yields 0 r exp in , where r and are cylindrical co- ∈ Z ψ ð Þ = rium systems (11), multiferroics (12), quantum systems (13), ion ordinates, n , and 0 0 0. This solution is called a vortex crystals (14, 15), and Bose–Einstein condensates (16) (the latter line, which is topologically equivalent to a string for the case two systems contain the effect of inhomogeneities due to, for = N 2 we discussed before. In the vicinity of the critical tem- example, temperature gradients). A detailed review concerning perature during a quench from the normal fluid to the superfluid the significance and limitations of these experiments can be state, ψ will be chosen randomly in uncorrelated regions, leading found in ref. 17. to a string network of normal fluid vortices. In condensed matter In this experimental study, we test the validity and applicability systems, the role of the limiting speed of light is taken by the of the Kibble–Zurek mechanism in a 2D colloidal model system sound velocity (in 4He, the second sound). This upper boundary whose equilibrium thermodynamics follow the microscopically leads to a finite speed of the propagation of order parameter motivated Kosterlitz–Thouless–Halperin–Nelson–Young (KTHNY) fluctuations and sets a “sonic horizon.” theory. This theory predicts a continuous, two-step melting be- Zurek argued that the correlation length is frozen-out close to havior whose dynamics, however, are quantitatively different the transition point or even far before depending on the cooling from phenomenological second-order phase transitions described rate (4, 5). Consider the divergence of the correlation length ξ for by the Ginzburg–Landau model. We applied cooling rates over ξ = ξ jej−ν e = ð − Þ= roughly three orders of magnitude, for which we changed the a second-order transition, e.g., 0 , where T Tc Tc is the reduced temperature. If the cooling is infinitely slow, the control parameter with high resolution and homogeneously system behaves as in equilibrium: ξ will diverge close to the throughout the sample without temperature gradients. Single transition and the system is a monodomain. For an instantaneous particle resolution provides a quantitative determination of de- quench, the system has minimal time to adapt to its surrounding: fect and domain structures during the entire quench procedure, and the precise knowledge of the equilibrium dynamics allows ξ will be frozen-out at the beginning of the quench. For second- determination of the scaling behavior of corresponding length order phase transitions, the divergence of correlation lengths is τ = τ jej−μ scales at the freeze-out times. In the following, we validate that accompanied by the divergence of the correlation time 0 , the Kibble–Zurek mechanism can be successfully applied to the which is due to the critical slowing down of order parameter KTHNY universality class. fluctuations. If the time t it takes to reach Tc for a given cooling rate is larger than the correlation time, the system stays in equi- Defects and Symmetry Breaking in 2D Crystallization librium and the dynamic is adiabatic. Nonetheless, for every finite The closed packed crystalline structure in two dimensions is a but nonzero cooling rate, t eventually becomes smaller than τ,and hexagonal crystal with sixfold symmetry. The thermodynamics of the system falls out of equilibrium before Tc is reached. The such a crystal can analytically be described via the KTHNY moment when both the correlation time and the time it takes to theory, a microscopic, two-step melting scenario (including two reach Tc coincide defines the freeze-out time. continuous transitions) that is based on elasticity theory and a
6926 | www.pnas.org/cgi/doi/10.1073/pnas.1500763112 Deutschländer et al. Downloaded by guest on September 29, 2021 orientation of the crystal axis. Spontaneous symmetry breaking
AB4 SEE COMMENTARY 5 3 implies that all possible global crystal orientations are degenerated, 4 2 -π/3 – 1 and the Kibble Zurek mechanism predicts that in the presence of 3 critical fluctuations the system cannot gain a global phase at finite 2 +π/3 19 18 cooling rates: Locally, symmetry broken domains will emerge, 1 17 which will have different orientations in causally separated regions. 14 16 The final state is a polycrystalline network with frozen-in defects. 4 13 As in the case of superfluid He, ψ ð~r , tÞ is complex with two 12 6 j independent components (N = 2). Consequently, we expect to ψ ð~ Þ observe monopoles in two dimensions. The phase of 6 rj, t is Fig. 2. Sketch of a fivefold oriented (A) and sevenfold oriented (B) dis- invariant under a change in the particular bond angles of clination. The red arrows illustrate the change in bond angle (blue) when ΔθjkðtÞ =±nπ=3(n ∈ N), which is caused by the sixfold orientation circling on an anticlockwise path around the defect. of the triangular lattice. Similar to the Higgs field or the superfluid, ψ ð~ Þ one cannot consider a closed (discrete) path in 6 r, t on which θjkðtÞ changes by an amount of ±π=3, leaving the orientational field renormalization group analysis of topological defects (18–20). In ψ ð~ Þ invariant. Reducing this path to a point, 6 r, t must tend toward the KTHNY formalism, the orientationally long range-ordered zero at the center to maintain continuity. Because the orientational crystalline phase melts at a temperature Tm via the dissociation field is defined at discrete positions, the defect is a single particle of pairs of dislocations into a hexatic fluid, which is unknown in marked as a monopole of the high symmetry phase. In fact, this 3D systems. This fluid is characterized by quasi-long-range ori- coincides with the definition of disclinations in the KTHNY for- entational but short-range translational order. In a triangular malism (20): The particle at the center is an isolated five- or sev- lattice, dislocations are point defects and consist of two neigh- enfold coordinated site. Fig. 2 illustrates this for a bond on a closed boring particles with five and seven nearest neighbors, re- path. Going counterclockwise around the defect, the bond angle spectively, surrounded by sixfold coordinated particles. At a changes by an amount of +π=3 for a fivefold (Fig. 2A)andby−π=3 higher temperature Ti, dislocations start to unbind further into for a sevenfold site (Fig. 2B). [In principle, also larger changes in isolated disclinations (a disclination is a particle with five or θjkðtÞ are possible, e.g., for n = 2, a four- or eightfold oriented site, seven nearest neighbors surrounded by sixfold coordinated par- but these are extremely rare.] In KTHNY theory the monopoles ticles), and the system enters an isotropic fluid with short-range (disclinations) combine to dipoles (dislocations) that can only an- orientational and translational order. A suitable orientational nihilate completely if their orientation is exactly antiparallel. At ψ ð~ Þ = orderP parameter is the local bond order field 6 rj, t − θ ð Þ Θ ð Þ finite cooling rates, they arrange in chains, separating symmetry 1 i6 jk t = ψ ð~ Þ i j t nj ke 6 rj, t e , which is a complex number with broken domains of different orientation: chains of dislocations can ψ ð~ Þ Θ ð Þ magnitude 6 rj, t and phase j t defined at the discrete par- be regarded as strings or 2D domain walls. ticle positions ~rj. ΘjðtÞ is the average bond orientation for a spe- cific particle. The k-sum runs over all nj nearest neighbors of Colloidal Monolayer and Cooling Procedure θ particle j, and jk is the angle of the kth bond with respect to a Our colloidal model system consists of polystyrene beads with certain reference axis. If particle j is perfectly sixfold coordinated diameter σ = 4.5 μm, dispersed in water and sterically stabilized θ ð Þ π= [e.g., all jk t equal an ascending multiple of 3], the local bond with the soap SDS. The beads are doped with iron oxide nano- ψ ð~ Þ = order parameter attains 6 rj, t 1. A five- or sevenfold co- particles that result in a superparamagnetic behavior and a mass ψ ð~ Þ J ordinated particle yields 6 rj, t 0. The three different density of 1.7 kg/dm3. The colloidal suspension is sealed within a ð Þ = phases can be distinguished via the spatial correlation g6 r millimeter-sized glass cell where sedimentation leads to the for- hψpð~Þψ ð!Þi ð Þ = hψ pð Þψ ð Þi 6 0 6 rj or temporal correlation g6 t 6 0 6 t of the mation of a monolayer of beads on the bottom glass plate. The local bond order parameter. For large r and t, respectively, each whole layer consists of >105 particles and in a 1,158 μm × 865 μm correlation attains a finite value in the (mono)crystalline phase, de- subwindow, ≈ 5,700 particles are tracked with a spatial resolution ∼ ð− =ξ Þ cays algebraically in the hexatic fluid, and exponentially exp r 6 of submicrometers and a time resolution in the order of seconds. ∼ ð− =τ Þ and exp t 6 in the isotropic fluid (20, 21). Unlike second- The system is kept at room temperature and exempt from density order phase transitions where correlations typically diverge alge- gradients due to a months-long precise control of the horizontal ξ τ braically, the orientational correlation length 6 and time 6 di- inclination down to microradian. The potential energy can be verge in the KTHNY formalism exponentially at Ti ξ ∼ jej−1=2 τ ∼ jej−1=2 [6] 6 exp a and 6 exp b , 6000 • τ 6000 Γ [1/s] 6 0.000042 fit where e = ðT − TiÞ=Ti, and a and b are constants (20, 22). This 5000 0.00037 5000 peculiarity is the reason why KTHNY melting is named contin- 0.0011 uous instead of second order. In equilibrium, the KTHNY sce- 4000 0.0023 4000 6
t 0.0060 nario has been verified successfully for our colloidal system in 3000 0.011 3000 various experimental studies (23–25). 0.020 To transfer this structural 2D phase behavior into the frame- 2000 0.033 2000 PHYSICS work of the Kibble–Zurek mechanism, we start in the high 1000 1000 temperature phase (isotropic fluid) and describe the symmetry breaking with the spatial distribution of the bond order parameter. 0 0 Because in 2D the local symmetry is sixfold in the crystal and the 30 35 40 45 50 55 60 65 fluid, the isotropic phase is a mixture of sixfold and equally num- Γ ( 1/Τ) bered five- and sevenfold particles (other coordination numbers Fig. 3. Orientational correlation time τ6 (experimental data and fit according are extremely rare and can be neglected). During cooling, isolated to Eq. 6) and the time t left until the transition temperature is reached for disclinations combine to dislocations that, for infinite slow cooling different cooling rates (colored straight lines, Eq. 9) as a function of inverse rates, can annihilate into sixfold particles with a uniform director temperature Γ (small Γ correspond to large temperatures and vice versa). The field. This uniformity is given by a global phase, characterizing the intersections define the freeze-out interaction values Γ^ = Γð^tÞ.
Deutschländer et al. PNAS | June 2, 2015 | vol. 112 | no. 22 | 6927 Downloaded by guest on September 29, 2021 isotropic fluid for independent equilibrium measurements. The 0.4 Γ Γ equilibrium i m measured values for the correlation time τ6 as well as a fit with
• 0.3 Γ [1/s] 0.033 −1=2 τ6 = τ0 exp bτj1=Γ − 1=Γcj , [8] ρ 0.020 0.2 0.011 0.0060 0.0023 is shown in Fig. 3 (right ordinate). The time left to the isotropic– 0.1 0.0011 hexatic transition is given by 0.00037 0.000042 0.0 _ t = ðΓi − ΓÞ Γ, [9] 12 and is also plotted in Fig. 3 (left ordinate) for various cooling rates 10 including the slowest and the fastest one. The points of intersection 2 0 8 h i ^ ^ _ −1=2 t = τ0 exp bτ 1 Γ t, Γ − 1=Γc , [10]
μ ðπ Þ3=2ð χ Þ2 Γ = Emag = 0 n H [7] , AB kBT 4πkBT
acts as inverse temperature (or dimensional pressure for fixed = = 2 volume and particle number). n 1 a0 is the 2D particle den- sity with a mean particle distance a0 ≈ 13 μm, and χ = 1.9 × 10−11 Am2=T is the magnetic susceptibility of the beads. Γ is the thermodynamic control parameter: A small magnetic field corre- sponds to a large temperature and vice versa. Measured values of the equilibrium melting temperatures are Γm ≈ 70.3 for the crystal/hexatic transition and Γi ≈ 67.3 for the hexatic/isotropic transition (25). The cooling procedure is as follows: we equili- brate the system deep in the isotropic liquid at Γ0 ≈ 25 and apply linear cooling rates Γ_ = ΔΓ=Δt deep into the crystalline phase up to C D Γend ≈ 100; thenceforward, we let the system equilibrate. We per- form different rates, ranging over almost three decades from Γ_ = 0.000042 1=suptoΓ_ = 0.0326 1=s. The slowest cooling rate corresponds to a quench time of ≈ 19 days and the fastest one to ≈ 40 min. We would like to emphasize that with the given control parameter there is no heat transport from the surface as in 3D bulk material. The lack of gradients rules out a temperature gradient-assisted annealing of defects that might be present in inhomogeneous systems. Structure and Dynamics of Defects and Domains The key element of the Kibble–Zurek mechanism is a frozen-out × μ 2 ≈ ^ Fig. 5. Snapshot sections of the colloidal ensemble (992 960 m , 4,000 correlation length ξ as the system falls out of equilibrium at the particles) illustrating the defect (A and C) and domain configurations (B and freeze-out time ^t. For slow cooling rates, the system can follow D) at the freeze out temperature Γ^ for the fastest (A and B: Γ_ = 0.0326 1=s, ^ _ ^ adiabatically closer to the transition (large ^ξ) than for fast rates Γ ≈ 30.3) and slowest cooling rate (C and D: Γ = 0.000042 1=s, Γ ≈ 66.8). The defects are marked as follows. Particles with the five nearest neighbors are where the systems reaches the freeze-out time earlier (small ^ξ). ^ red, seven nearest neighbors are green, and other defects are blue. Sixfold To find t, we determine the orientational correlation time τ6 coordinated particles are gray. Different symmetry broken domains are col- according the KTHNY theory by fitting g6ðtÞ ∼ expð−t=τ6Þ in the ored individually, and high symmetry particles are displayed by smaller circles.
6928 | www.pnas.org/cgi/doi/10.1073/pnas.1500763112 Deutschländer et al. Downloaded by guest on September 29, 2021 and ξ = 1.56 ± 0.03. For the slowest cooling rate Γ_ = AB dom 0.000042 1=s(Fig.5C and D)whereΓ^ = 66.7, the mean dis- SEE COMMENTARY tance between defects, as well as the typical domain size, is signif- icantly larger compared with the fastest cooling rate. We observe ξ = ± ξ = ± def 2.36 0.07 and dom 2.30 0.09. To allow relaxation of the defect and domain structure after the freeze-out time (6), we keep the temperature constant after Γend ≈ 100 is reached. Fig. 6 shows the defect and domain config- urations after an equilibration time of ≈ 5 h for the fastest cooling rate (Fig. 6 A and B) where the quench time was ≈ 40 min and after an equilibration time of ≈ 3 days for the slowest rate (Fig. 6 C and D) where the quench time was ≈ 19 days. The different length C D scale of the defect and symmetry broken domain network in re- spect to the cooling rate is clearly visible: although we observe a large number of domains for fast cooling, slow cooling results in merely two large domains separated by a single grain boundary. The final evolution will be given by classical coarse graining. The ground state is known to be a monodomain but its observation lies beyond experimental accessible times for our system. Scaling Behavior The main prediction of the Kibble–Zurek mechanism is a power law dependence of the frozen-out correlation length ^ξ as a function of τq (Eq. 5), which results from the algebraic di- vergence of the correlation, presuming Eqs. 3 and 4. In KTHNY Fig. 6. Snapshot sections of the colloidal ensemble illustrating the defect ξ τ A C B D melting, 6 and 6 diverge exponentially, and one has to solve Eq. ( and ) and domain configurations ( and ) after quasi-equilibration of the 10 ^ðΓ_ Þ system for the fastest (A and B: Γ_ = 0.0326 1=s, Γ ≈ 105) and slowest cooling to find the implicit dependency t . We did this numerically end −5 _ −2 rate (C and D: Γ_ = 0.000042 1=s, Γ ≈ 98). The system size and the labeling of for discrete values in the range of 4 × 10 ≤ Γ ≤ 4 × 10 and end ^ξ defects and domains are the same as in Fig. 5. determined the frozen-out orientational correlation length 6 for τ =τ = ξz a scaling 6 B c 6 with the dynamical exponent z (22). Here, τB ≈ 171.6 s is the Brownian time that is the time a single particle The domain structure, on the other hand, can be characterized needs to diffuse its own diameter. Using Eq. 8, one finds with quantitatively by analyzing symmetry broken domains with a Γðt, Γ_ Þ from Eq. 9 the expression ψ ð~ Þ similar phase of 6 rj, t . According to ref. 26, we define a particle " − = # to be part of a symmetry broken domain if the following three 1=z _^ _ 1 2 τ bτ Γ − Γ + Γt Γ conditions are fulfilled for the particle itself and at least one ^ξ Γ_ = 0 c i [11] 6 exp _^ _ . ψ ð~ Þ cτB z Γ Γ − Γ Γt Γ nearest neighbor: (i) the magnitude 6 rj, t of the local bond c i c order parameter must exceed 0.6 for both neighboring particles; (ii) the bond length deviation of neighboring particles is less than This function is plotted in Fig. 7 for z = 4.5 and c = 0.83 (red ξ 10% of the average particle distance a0 ;and(iii) the variation in curves) on a double logarithmic scale together with def and ΔΘ ð Þ = ½ψ ð~Þ − ½ψ ð~Þ the average bond orientation ij t Im 6 ri Im 6 rj of neighboring particles i and j must be less than 14° (less than 14°=6 in real space). Simply connected domains of particles that fulfill 2.4 • ~ Γ -0.061(1) all three criteria are merged to a local symmetry broken domain.
If a particle does not satisfy these conditions in respect to a def neighboring particle, it is assigned to the high symmetry phase ξ 2.0 [almost all defects are identified as such due to their small value ψ ð~ Þ of 6 rj, t ]. Fig. 4 (Lower) shows the evolution of the ensemble average domain size hAi as a function of Γ. We observe a behavior analog to ρ: domain formation significantly deviates from the 1.6 Γ equilibrium case before i, namely around the freeze-out tem- 2.4 perature Γ^ of the corresponding cooling rates Γ_. To compare both ~ Γ• -0.061(1) networks in the following, we define the dimensionless lengths −1=2 2 1=2 dom
ξ = ρ ξ = ðh i= Þ ξ def and dom A a0 , which display the characteris- 2.0 tic length scales in units of a0. Colloidal ensembles offer the unique possibility to monitor the system and its domain and defect structure on single particle PHYSICS level. Fig. 5 illustrates both (Left column for defects and right 1.6 column for domains) at the freeze-out temperature Γ^ for the 10-4 10-3 10-2 fastest (Fig. 5 A and B) and the slowest (Fig. 5 C and D) cooling • rate. For Γ_ = 0.0326 1=s (Fig. 5 A and B) where ^t is already Γ reached at Γ^ = 30.3, the defect density is large, as is the number ξ ξ of high symmetry particles. However, there is a significant Fig. 7. Length scale of the defect def and domain network dom is plotted as a function of the cooling rate Γ_ (open symbols). Red lines are numerical number of sixfold coordinated particles and a few orientationally solutions of the transcendental equation following the freeze-out condition ordered domains (to accord for finite size effects, we will exclude for the KTHNY-like divergences (see text for definition). For comparison, domains that hit the border of the field of view when evaluating dashed blue lines are power law fits predicted by the standard Kibble–Zurek ξ Γ^ ξ = ± κ ≈ ξ ξ dom at ). At this point, the length scales are def 1.56 0.01 mechanism that show the same algebraic exponent 0.06 for def and dom.
Deutschländer et al. PNAS | June 2, 2015 | vol. 112 | no. 22 | 6929 Downloaded by guest on September 29, 2021 ξ Γ^ dom at the freeze-out temperature . We find very good agree- single particle resolution. The theoretical framework is given by the ment. Nonetheless, we fit for comparison the data via an alge- Kibble–Zurek mechanism, which describes domain formation on _ _ −κ braic scaling (blue dotted lines) of the form fðΓÞ = aΓ , for different scales like the Higgs field shortly after the Big Bang or the κ = ± κ = ± 4 which we observe def 0.061 0.001 and dom 0.061 0.002. vortex network in He quenched into the superfluid state. Along The data are compatible with the algebraic decay only for in- various cooling rates, we analyzed the development of defects and termediate cooling rates. The deviations from the standard Kib- – symmetry broken domains when the systems fall out of equilibrium ble Zurek mechanism for systems with second-order transitions and fluctuations of the order parameter cannot follow adiabatically are in line with the temperature-quenched 2D XY model (27) due to critical slowing down. Although 2D melting in the colloidal also having nonalgebraic divergences of the correlation length in equilibrium and thus being in a similar universality class. The monolayer is described by KTHNY theory where the divergence of small algebraic exponent κ can be explained by the relatively the relevant correlation lengths in equilibrium is exponential large value of the dynamical exponent z, which regulates the (rather than algebraic as typically found in 3D systems), the central ^ξ ðΓ_ Þ = idea of the Kibble–Zurek mechanism still holds, and the scaling of slope of 6 (in ref. 22, a value z 2.5 was proposed for the hard-disk system). The large value of z is due to quite long the observed domain network is correctly described. Implicitly, this correlation times in this colloidal system (Fig. 3), which are caused shows that existence of grain boundaries cannot solely be used as by its overdamped dynamics. Note that the sonic horizon is set by criterion for first-order phase transitions and nucleation or to the sound velocity of the colloidal monolayer (and not the solvent) falsify second/continuous-order transitions because they natu- being approximately millimeters per second, which is six orders of rally arise for nonzero cooling rates. Those will always be present magnitude slower compared with atomic systems. on finite time scales in experiments and computer simulations after preparation of the system. Conclusions We presented a colloidal model system, where structure formation ACKNOWLEDGMENTS. P.K. thanks Sébastien Balibar for fruitful discussion. P.K. in spontaneously symmetry broken systems can be investigated with received financial support from German Research Foundation Grant KE 1168/8-1.
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6930 | www.pnas.org/cgi/doi/10.1073/pnas.1500763112 Deutschländer et al. Downloaded by guest on September 29, 2021